# Does an electron have kinetic energy when attached to a proton?

• B
• Lolicon
That is, if we could measure the kinetic energy of all the electrons in the box at every moment, we would get a number that is not zero but is smaller than the total energy of the system.The expected value of kinetic energy is smaller than the total energy of the system.f

#### Lolicon

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?
A proton and electron bound together is called a hydrogen atom. The hydrogen atom has a set of possible energies, each of which is the sum of the kinetic energy and potential energy.

vanhees71 and Dale
Yes, it has KE

There is a nice result called the Virial theorem which says that if the potential energy of interaction between two bound particles goes as ##V(r) = \lambda r^n##, then the potential and kinetic energies of that system are related via ##2 \langle T \rangle = n \langle V \rangle ##. For an electrostatic interaction that potential has ##n=-1##, so ##2\langle T \rangle = - \langle V \rangle##. Equivalently, the total energy is ##E = \langle T \rangle + \langle V \rangle = - \langle T \rangle = \langle V \rangle/2##.

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##\langle V \rangle = -27.2 \text{eV}## and ##\langle T \rangle = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

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Lolicon, JD_PM, berkeman and 1 other person
The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##V = -27.2 \text{eV}## and ##T = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

Those are, of course, the expected values of ##T## and ##V## for a QM system.

JD_PM and etotheipi
Those are, of course, the expected values of ##T## and ##V## for a QM system.

Thanks, give me two seconds and I'll put the langles and the rangles where they belong

JD_PM and vanhees71
Possible nitpick about terminology: "expectation value" is the term that I've always seen in English-language QM textbooks, journal articles, etc., at least in the US.

To me, "expected value" has the connotation "It has a definite value, and this is what I expect it to be." Of course, that's not what we're talking about here.

weirdoguy and etotheipi
Whenever an electron is moving in some limited space, as when orbiting an atom or put in a "box", it has a nonzero expectation value of kinetic energy.